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Properties

Label 2-570-285.284-c1-0-22
Degree $2$
Conductor $570$
Sign $0.766 + 0.642i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−0.5 + 1.65i)3-s − 4-s + (1.91 + 1.15i)5-s + (1.65 + 0.5i)6-s − 3.21i·7-s + i·8-s + (−2.5 − 1.65i)9-s + (1.15 − 1.91i)10-s − 4.31i·11-s + (0.5 − 1.65i)12-s + 3.31·13-s − 3.21·14-s + (−2.87 + 2.59i)15-s + 16-s + 3.21·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 + 0.957i)3-s − 0.5·4-s + (0.855 + 0.518i)5-s + (0.677 + 0.204i)6-s − 1.21i·7-s + 0.353i·8-s + (−0.833 − 0.552i)9-s + (0.366 − 0.604i)10-s − 1.30i·11-s + (0.144 − 0.478i)12-s + 0.919·13-s − 0.860·14-s + (−0.742 + 0.669i)15-s + 0.250·16-s + 0.780·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37366 - 0.499860i\)
\(L(\frac12)\) \(\approx\) \(1.37366 - 0.499860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.5 - 1.65i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
19 \( 1 + (4.31 - 0.605i)T \)
good7 \( 1 + 3.21iT - 7T^{2} \)
11 \( 1 + 4.31iT - 11T^{2} \)
13 \( 1 - 3.31T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
23 \( 1 - 7.04T + 23T^{2} \)
29 \( 1 - 8.25T + 29T^{2} \)
31 \( 1 - 2.61iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 3.82iT - 43T^{2} \)
47 \( 1 - 8.86T + 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 1.81iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + 6.43T + 83T^{2} \)
89 \( 1 - 2.61T + 89T^{2} \)
97 \( 1 + 3.36T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61040898467657513258625814788, −10.22173307659288407374285577723, −9.058871093426804019965927558579, −8.400901340166381553949305063903, −6.79898497328661438223456341823, −5.92591415126299207618053038508, −4.90734183302549374764122157878, −3.68887578599870942221811668942, −3.04632443277903002603449380817, −1.03292326427899549499258678015, 1.43489400938269139052240448467, 2.65190830678613713680380445554, 4.73958553200189097475567714687, 5.51369817387461804200516101872, 6.30462141542472313279751005870, 7.02381065053566138576728950320, 8.310610177631154624861490462099, 8.774360322781266351555593450392, 9.727366095426962241855385281793, 10.81024435039304268921233982717

Graph of the $Z$-function along the critical line